no. 1
Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations
Annales de l'Institut Fourier, Tome 32 (1982) no. 1, pp. 39-69

Let G be a locally compact group. Let L t be the left translation in L ∞ (G), given by L t f(x)=f(tx). We characterize (undre a mild set-theoretical hypothesis) the functions f∈L ∞ (G) such that the map t→L t f from G into L ∞ (G) is scalarly measurable (i.e. for ϕ∈L ∞ (G) * , t→ϕ(L t f) is measurable). We show that it is the case when t→θ(L f t) is measurable for each character θ, and if G is compact, if and only if f is Riemann-measurable. We show that t→L t f is Borel measurable if and only if f is left uniformly continuous.

Some of the measure-theoretic tools used there have independent interest. For example, if a set of measurable functions on [0,1] is separable and point-wise relatively compact, the same is true of its convex hull.

Soit G un groupe localement compact. Soit L t la translation à gauche dans L ∞ (G) donnée par L t f(x)=f(tx). On caractérise (sous des axiomes peu restrictifs de théorie des ensembles) les f∈L ∞ (G) telles que l’application t→L t f de G dans L ∞ (G) soit scalairement mesurable (c’est-à-dire que t→ϕ(L t f) est mesurable pour ϕ∈L ∞ (G) * ). On montre que c’est le cas dès que pour tout caractère θ de L(G), t→θ(L f t) est mesurable, et dans le cas compact, cela caractérise les fonctions Riemann-mesurables. On montre que l’image réciproque de tout borélien de L ∞ (G) par l’application t→L t f est mesurable si et seulement si f est uniformément continue.

Les outils de théorie de la mesure utilisés ont un intérêt en soi. Par exemple un ensemble de fonctions mesurables sur [0,1] est séparable et relativement compact pour la topologie de la convergence ponctuelle, il en est de même de son enveloppe convexe.

Talagrand, Michel. Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations. Annales de l'Institut Fourier, Tome 32 (1982) no. 1, pp. 39-69. doi: 10.5802/aif.859
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